Endpoint sufficiency at the upper weighted boundary

Determine whether the metric-entropy condition \(\sup_{0<\delta<1}N(E,\delta)\delta^{(d-1)(p-1)-\alpha}<\infty\) is sufficient for boundedness of the local spherical maximal operator \(M_E\) on \(L^p(|x|^\alpha)\) at the upper endpoint \(\alpha=U(p)\), for \(p\ge 1+\beta/(d-1)\), including the case \(\alpha=0\) and \(p=1+\beta/(d-1)\).

Background

For a dilation set E⊂[1,2]E\subset[1,2], the paper derives a necessary upper-boundary condition for weighted boundedness by testing the maximal operator on characteristic functions of thin annuli. In the notation of the paper, this condition is sup⁡0<δ<1N(E,δ)δ(d−1)(p−1)−α<∞\sup_{0<\delta<1}N(E,\delta)\delta^{(d-1)(p-1)-\alpha}<\infty, and the corresponding boundary is α=U(p)\alpha=U(p).

The authors explicitly state that it is unknown whether this necessary metric-entropy condition becomes sufficient at the endpoint. They identify the unweighted critical case α=0\alpha=0, p=1+β/(d−1)p=1+\beta/(d-1), as a particular unresolved instance.

References

One might ask whether for $p\ge 1+\frac\beta{d-1}$ the condition $\sup_{0<\delta<1} N(E,\delta)\delta{(d-1)(p-1) -\alpha}<\infty$ is also sufficient for $Lp(w_\alpha)$-boundedness when $\alpha=U(p)$ but this is open (in particular for $\alpha=0$, $p=1+\frac\beta{d-1}$).

— Endpoint power weight inequalities for spherical maximal functions  (2610.18316 - Fraccaroli et al., 13 Oct 2026) in Introduction, subsection 'Two necessary conditions'

The behavior of $M_{E\mu}$ on $L{p(\mu)} (w_\alpha)$ remains open for $\alpha=1-d+(\mu+1){-1}$.

— Endpoint power weight inequalities for spherical maximal functions  (2610.18316 - Fraccaroli et al., 13 Oct 2026) in Introduction, immediately after Theorem (sequence examples)